Approximate the mean for the following grouped frequency distribution table:
| Data | Frequency |
| :------ | :-------- |
| 50-54 | 12 |
| 55-59 | 20 |
| 60-64 | 11 |
| 65-69 | 10 |
| 70-74 | 7 |
| 75-79 | 4 |
| 80-84 | 2 |
| 85-89 | 1 |
| 90-94 | 1 |
mean = $\square$
Enter an integer or decimal number.
Report answer accurate to one decimal place.
Answer (2)
Calculate the midpoint of each interval.
Multiply each midpoint by its corresponding frequency.
Sum the products and divide by the total frequency: 68 4271 ≈ 62.8 .
Round the result to one decimal place. 62.8
Explanation
Understand the problem We are given a grouped frequency distribution table (GFDT) and asked to approximate the mean. The data is grouped into intervals, and we have the frequency of observations within each interval.
Find the midpoints of the intervals To approximate the mean, we first find the midpoint of each interval. The midpoint is calculated as the average of the lower and upper limits of the interval.
Calculate weighted values Next, we multiply the midpoint of each interval by its corresponding frequency. This gives us the weighted value for each interval.
Calculate the approximate mean Then, we sum all the weighted values and divide by the sum of the frequencies. This gives us the approximate mean of the data.
Calculations and final answer The intervals and their corresponding frequencies are:
50 − 54 : 12 55 − 59 : 20 60 − 64 : 11 65 − 69 : 10 70 − 74 : 7 75 − 79 : 4 80 − 84 : 2 85 − 89 : 1 90 − 94 : 1
The midpoints are:
2 50 + 54 = 52 2 55 + 59 = 57 2 60 + 64 = 62 2 65 + 69 = 67 2 70 + 74 = 72 2 75 + 79 = 77 2 80 + 84 = 82 2 85 + 89 = 87 2 90 + 94 = 92
The weighted values are:
52 × 12 = 624 57 × 20 = 1140 62 × 11 = 682 67 × 10 = 670 72 × 7 = 504 77 × 4 = 308 82 × 2 = 164 87 × 1 = 87 92 × 1 = 92
The sum of the weighted values is:
624 + 1140 + 682 + 670 + 504 + 308 + 164 + 87 + 92 = 4271
The sum of the frequencies is:
12 + 20 + 11 + 10 + 7 + 4 + 2 + 1 + 1 = 68
The approximate mean is:
68 4271 ≈ 62.808823529411766
Rounding to one decimal place, we get 62.8 .
Examples
In environmental science, you might collect data on air pollution levels across different areas. Each area's pollution level falls into a certain range (like 10-20 ppm of a pollutant), and you record how many areas fall into each range. By calculating the mean of this grouped data, you can estimate the average pollution level across all the areas you studied. This helps in understanding the overall environmental quality and identifying areas that need more attention.
To calculate the mean of the grouped data, we find the midpoints of each interval, multiply them by their frequencies, then sum those products and divide by the total frequency. The mean is approximately 62.8 when rounded to one decimal place. This provides an estimate of the average of the data represented in the frequency distribution.
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